Cooperative Amplify-and-Forward Relaying Systems with Quadrature Spatial Modulation

Transcription

1 Cooperative Amplify-and-Forward Relaying Systems with Quadrature Spatial Modulation IBRAHEM E. ATAWI University of Tabuk Electrical Engineering Department P.O.Box:74, 749 Tabuk SAUDI ARABIA Abstract: Quadrature spatial modulation QSM is a recent digital multiple-input multiple-output MIMO transmission technique. Combined with cooperative relaying, QSM improves the overall spectral efficiency and enhances the communication reliability. In this paper, we study the performance of QSM amplify-and forward cooperative relaying systems. In particular, a closed-form expression for the average pair-wise error probability PEP of the cooperative system is derived, which is used to calculate a tight upper bound of the average bit error probability ABEP over Rayleigh fading. In addition, a simple approximate, yet accurate expression is derived and analyzed asymptotically. Simulation results, which corroborate the numerical ones, show the effectiveness of combined QSM and cooperative relaying in improving the overall system performance. Key Words: Amplify-and-forward, Quadrature spatial modulation, Spectral efficiency, MIMO. Introduction The unprecedented demands for huge data rates and high speed wireless communication applications motivated both researchers and industry to investigate several recent technologies, which led to many promising innovations. Such innovations are expected to support many critical services that affect all aspects of human lives. Quadrature Spatial modulation QSM is proposed as a promising technology for the future multiple-input multiple-output MIMO wireless networks []. It has been shown that QSM increases the spectral efficiency of conventional spatial modulation SM system while retaining all SM inherent advantages. The techniques of QSM, SM and spaceshift-keying SSK are introduced as low-complexity and spectral-efficient implementations of MIMO systems [], [3], [4]. In addition, they avoid conventional MIMO drawbacks including inter-channel interference ICI and complex receivers []- [5]. In QSM, the spatial constellation symbols are extended to orthogonal in-phase and quadrature components. One component transmits the real part of a signal constellation symbol and the other transmits the imaginary part. It is worth pointing that, in conventional SM, these two parts are transmitted from a single transmit antenna to avoid ICI at the receiver input. However, in a QSM system, ICI is also avoided entirely since the two transmitted data are orthogonal and modulated on the real part and the imaginary part of the carrier signal. Very recently, few works studied QSM in conventional MIMO system assuming perfect and imperfect channel state information CSI [], [5], [6]. Authors in [7] proposed Bi-SSK system, which is considered as a special case of QSM without using any modulation schemes. Although QSM has been studied in point-to-point MIMO systems, it is not yet investigated in the context of cooperative systems. As such, this paper studies a QSM single amplify-and-forward AF relaying system and analyzes the system error performance. In this work, the average bit error probability ABEP for QSM-AF cooperative systems is analyzed. In particular, a closed-form expression for the average pairwise error probability PEP is derived employing the optimal maximum likelihood ML detector at the receiver. Based on the derived PEP expression, a tight upper bound expression is obtained using the union bound formula. Moreover, an asymptotic analysis is performed to get insights on key parameters. The results show the effectiveness of cooperative QSM in improving the error system performance. The remainder of this paper is organized as follow: Section describes system and channel models. Performance analysis is presented in Section 3. E-ISSN: Volume 5, 6

2 Numerical results are presented in Section 4. Finally, Section 5 concludes the paper. System Model. Channel Model We consider a system model comprising a source S with N t transmit antennas, a single antenna AF relay R, and a single antenna destination D as depicted in Fig. []. The cooperative system is operating over Rayleigh fading channels. We assume that b = Figure : System model of a single RF chain QSM-AF system. log MNt incoming bit stream enters the source at each transmission instant. The incoming data bits are processed and partitioned into three groups. S determines the index of the active transmit antennas by using two groups of log N t bits ofb, then maps the remaining log M bits onto the corresponding M-ary quadrature amplitude modulation M-QAM/ phase shift keying M-K or other complex signal constellation diagrams. The signal constellation symbol, x, is further decomposed to its real, x β, and imaginary, x b, parts. The real part is transmitted from one transmit antenna among the existing N t transmit antennas, where the active antenna index is determined by the first log N t bits. Similarly, the imaginary part is transmitted by another or the same transmit antenna depending on the other log N t bits. However, the transmitted real and imaginary parts are orthogonal representing the in-phase and the quadrature components of the carrier signal. An example for QSM bits mapping and transmission is given in what follows assuming N r N t,4 4-MIMO system and 4-QAM modulation. The number of data bits that can be transmitted at one particular time instant is b = log Nt M = 6 bits. [ Assume that the following ] incoming data bits, b = are to be transmitted. The first }{{} log M+log Nt log M bits [ ], modulate a 4-QAM symbol, x = +j. This symbol is divided further into real and imaginary parts, x β = and x b =. The second log N t bits [ ], modulate the active antenna index, l β = 3 to transmit x β = resulting in the transmitted vector s β = [ ] T. The last log N t bits, [ ], modulate the active antenna index,l b = 4, used to transmitx b =, resulting in the vector s b = [ ] T. The transmitted vector is then obtained by adding the real and imaginary vectors, s = s β +js b = [ +j ] T. The transmission protocol consists of two time slots. In the first time slot, the vector, s, is transmitted to the relay, over ann t Rayleigh fading wireless channel. In the second time slot, the AF relay forward the received signal to the destination. In the first phase, the received signal at R can be written as y s,r = h αx β +j h ax b +η r, α,a =,,,N t ;β,b =,..,M; where x β and x b are symbols in the PAM signalconstellation diagram and h α and h a are the activated antenna fading gains, are assumed to be complex Gaussian random variables with zero mean and variances σ h, and η r CN, is the complex Gaussian noise with zero mean and variance. In the second phase, in AF relaying, the relayed signal x R is an amplified version of the input signal at the relay node, i.e., x R = A y s,r, where A is the amplification factor. Using this technique the amplification process is performed in the analogue domain and consists of a simple normalization of the total received power without further processing. Hence, the received signal at D from R in the second phase can be written as y r,d = y s,r +n r,d Then, = h αx β +j h ax b +η r +η r,d = h αx β +j h ax b +η r +n r,d }{{}}{{} Noise Part Signal Part y r,d = A g + h αx β +j A g + h ax b + ˆn, 3 where g is the fading gain between R and D, A =, and ˆn is Gaussian noise with variance. σ h /+ E-ISSN: Volume 5, 6

3 Since the channel inputs are assumed equally likely, the optimal detector, based on the ML principle, is given as ] [ĥα,ĥa,ˆx β,ˆx b = arg min α,a,β,b y r,d A g + [h αx β +jh a x b ] = arg min α,a,β,b λ R { y H λ }, 4 whereλ = A g + [h α x β +jh a x b ]. It can be seen that optimal detection requires a joint detection of the antenna indices and symbols. 3 QSM-AF Performance Analysis In this section, the ABEP is investigated, where a closed-form expression for the average PEP is derived. From which we derive a tight upper bound ABEP expression. 3. SNR statistics Assuming λ = A g + [h α x β +jh a x b ] is transmitted, the probability of deciding in favour of ˆλ = WSEAS TRANSACTIONS on COMMUNICATIONS A g + [ĥαˆx β +jĥaˆx b ] is given as Pr λ ˆλ H = Pr d λ > dˆλ H = Q χ 5 where d λ = λ R { y H λ }. We define χ as χ λ N ˆλ = A g A g + where j = and 4 C +jd, C = h R α x β h I a x b ĥrαˆx β +ĥiaˆx b D = h I α x β +h R a x b ĥiαˆx β ĥr a ˆx b Since the symbols x β and x a are drawn from a real constellation, i.e., PAM, C and D are independent. Therefore, χ has the following mean since h a,h α and g are independent Q A χ = Q g A g C +jd +4N P R g = Q C +jd P R g + P R 4N A Λ = Q C +jd Λ+Ξ4N, 9 where Λ = P R g, and Ξ = P R. A Note that ν = 4 C + jd is an exponential random variable with the following mean 4 κ if h α ĥα,h a ĥa, 4 κ if h α = ĥα,h a ĥa, 4 κ 3 if h α ĥα,h a = ĥa, 4 κ 4 if h α = ĥα,h a = ĥa, where κ = x β + ˆx β + x b + ˆx b, κ = x β ˆx β + x b + ˆx b, κ 3 = x β + ˆx β + x b ˆx b, and κ 4 = x β ˆx β + x b ˆx b. In the case that we have BK, i.e.±, ζ can be greatly simplified to if h α ĥα,h a ĥa, 4N x β ˆx β + if h α = ĥα,h a ĥa, 4 + x b ˆx b if h α ĥα,h a = ĥa, 4 κ 4 if h α = ĥα,h a = ĥa, and the terms x b ˆx b and x β ˆx β can be either or 4. Furthermore, if we assume Quadrature space shift keying QSSk, i.e., x β = x b =, ζ can be simplified to { if h α ĥα,h a ĥa, σ h if h α = ĥα,h a ĥa, or h α ĥα,h a = ĥa To find the average error probability, we need to find the CDF of χ. This can be done as follows: The PDF of Λ is f Λ x = Λ exp x Λ, where Λ = σ g P R/, and the PDF of the termν = 4 C+ E-ISSN: Volume 5, 6

4 jd is f ν x = ν exp x/ ν. Therefor CDF of χ = Λ Λ+Ξ 4 C +jd is derived as Λ F χ x = Prχ < x = Pr C +jd < x Λ+Ξ4 Ξx = exp x ν Ξx K, 3 wherek v. is thev th -order modified Bessel function of the second kind. 3. Exact Average PEP The alternative expression of the average pairwise error probability can be written as PEP = a b exp bx F χ xdx π x 4 Substituting 3 into 4 and solve the integral with the help of [9, ,pp.698] we have a =,b = PEP = Ξ ν Λ+ ν + ν exp Ξ Λ+ ν ] Ξ Ξ [K K 5 Λ+ ν Λ+ ν 3.3 ABEP Performance After the evaluation of the average PEP, the ABEP of the proposed scheme can be upper bounded by the following asymptotically tight union bound: P b = k k k n=m= k Prλ n λ m e n,m, 6 where {λ n } k n= is the set of all possible QSM symbols, k = log MNt is the number of information bits per QSM symbol, and e n,m is the number of bit errors associated with the corresponding PEP event. Note that M represents the PAM constellation. 3.4 High SNR Analysis Although the expressions for the average error probability in 5 enable numerical evaluation of the system performance and may not be computationally intensive, they do not offer insight into the effect of the system parameters. We now aim at expressing F χ x ABEP 3 4 QSM, Analytical QSM, Asymptotic QSM, Simulations SM, Simulations E /N [db] t Figure : Analytical, simulation and asymptotic ABEP vs. Et db with N t =, and 4-QAM scheme. and average PEP in simpler forms. This will ease the analysis of the optimization problems. The following steps will be done to simplify the expression of the error probability: According to [] and [], the asymptotic error and outage probabilities can be derived based on the behavior of the CDF of χ around the origin. By using Taylor s series, F χ x can be rewritten as F χ x ν + Ξ ψ log Ξ x+h.o.t 7 whereψ. is the digamma function, note thatψ =.577. Substituting 7 into 4 and solve the integration we have PEP ν + Ξ Ξ ψ log. 8 It is noted, in the Numerical Results Section, that this approximate error expression is in an excellent agreement with the exact expression especially in the pragmatic SNR values. 4 Numerical analysis and Discussion In this section, the performance of the QSM singe AF cooperative relaying system is evaluated via analytical results and validated through simulations for 4-QAM scheme. Unless otherwise stated, we assume N t =, =,N o =, and E t = Es+E r. For comparison purposes, the performance of SM is included [8]. In Fig., the ABEP versus Et is evaluated and simulated for N t =,4, respectively. It can be observed that the ABEP performance improves with increasing SNR. In the high Et region, the asymptotic E-ISSN: Volume 5, 6

5 ABEP performance becomes in excellent agreement with exact one. The performance of conventional SM scheme with 8-QAM modulation achieving similar spectral efficiency 4-QAM in the considered QSM system is depicted in both figures. Obtained results demonstrate the significant enhancement of the QSM system over the SM system, where a gain of about 3 db can be noticed in the figures. Note that this gain is attained at almost no cost. The receiver complexity of QSM and SM schemes are equivalent and depends on the considered spectral efficiency as reported in []. 5 Conclusion We analyzed the performance of the QSM-AF cooperative systems employing ML detector at the receiver. The tight upper bounded ABEP was derived using the closed-form average PEP. In addition, the asymptotic performance analysis was performed; a simple approximate error expression was derived at high SNR values. The QSM cooperative system outperforms the conventional SM cooperative system beside achieving higher spectral efficiency and maintaining most of its inherit advantages without any additional receiver complexity. References: [] R. Mesleh, S. Ikki, and H. goune, Quadrature spatial modulation, IEEE Trans. on Veh. Tech.,, vol. 64, no. 6, pp , June 5. [] M. Di Renzo, H. Haas, A. Ghrayeb, S. Sugiura, and L. Hanzo, Spatial modulation for generalized MIMO: Challenges, opportunities, and implementation, Proc. of the IEEE, vol., no., pp. 56 3, Jan 4. [3] J. Jeganathan, A. Ghrayeb, L. Szczecinski, and A. Ceron, Space shift keying modulation for MIMO channels, IEEE Trans. Wireless Commun., vol. 8, no. 7, pp , Jul. 9. [4] R. Mesleh, H. Haas, S. Sinanović, C. W. Ahn, and S. Yun, Spatial modulation, IEEE Trans. on Veh. Tech., vol. 57, no. 4, pp. 8 4, Jul. 8. [5] R. Mesleh and S. Ikki, On the impact of imperfect channel knowledge on the performance of quadrature spatial modulation, in Proc. IEEE Wireless Communications and Networking Conference WCNC, pp , 9- March 5. [6] R. Mesleh and S. Ikki, A high spectral efficiency spatial modulation technique, in Proc. IEEE Vehicular Technology Conference VTC Fall, pp. 5, 4-7 Sept. 4. [7] L. Han-Wen, R.Y. Chang, C. Wei-Ho, H. Zhang, and K. Sy-Yen, Bi-space shift keying modulation for MIMO systems, IEEE Communications Letters, vol. 6, no. 8, pp. 6 64, August. [8] R. Mesleh, S. Ikki, and M. Alwakeel, Performance analysis of space shift keying with amplify and forward relaying, IEEE Communications Letters, vol. 5, no., pp , December. [9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. Elsevier, 7. in a fading environment when using multiple antennas, Wireless Personal Communications, vol. 6, no. 6, pp , 998. [] Z. Wang and G. B. Giannakis, A simple and general parameterization quantifying performance in fading channels, IEEE Trans. on Commun., vol. 5, no. 8, pp , Aug. 3. [] A. Ribeiro, X. Cai, and G. B. Giannakis, Symbol error probabilities for general cooperative links, IEEE Trans. Wireless Commun., vol. 4, no. 3, pp , May 5. E-ISSN: Volume 5, 6